Title:

United States Patent 6836448

Abstract:

A system and method of creating a filter for use with locally dense seismic data is disclosed. The method includes obtaining survey geometry characteristics from a locally dense seismic survey. A filter is designed which uses spatial derivatives of the wavefield of order between (1) and the maximum order of spatial derivatives of the wavefield that can be estimated within a group. The filter can be designed so as to separate up/down going components, p/s components, or both up/down and p/s components. Partial derivatives in space and time of the wavefield can be calculated, using, for example, a taylor series expansion as an approximation. The seismic data is filtered by combining estimated near surface material properties, the seismic data, and the calculated partial derivatives.

Inventors:

Robertsson, Johan Olof Anders (Oslo, NO)

Curtis, Andrew (Edinburgh, GB)

Curtis, Andrew (Edinburgh, GB)

Application Number:

10/181232

Publication Date:

12/28/2004

Filing Date:

07/09/2002

Export Citation:

Assignee:

Schlumberger Technology Corporation (Ridgefield, CT)

Primary Class:

Other Classes:

367/21, 367/38, 367/43, 367/58, 702/17

International Classes:

Field of Search:

367/43, 367/24, 367/45, 367/44, 702/14, 367/38, 702/17, 367/58, 367/21, 367/59, 367/46

View Patent Images:

US Patent References:

4979150 | System for attenuation of water-column reverberations | 1990-12-18 | Barr | |

4935903 | Reinforcement of surface seismic wavefields | 1990-06-19 | Sanders et al. | 367/24 |

4870580 | Compressional/shear wave separation in vertical seismic profiling | 1989-09-26 | Lang et al. | 702/17 |

4648039 | Compressional/shear wave separation in vertical seismic profiling | 1987-03-03 | Devaney et al. | 702/17 |

Foreign References:

EP0327758 | 1989-08-16 | Process for separating upgoing and downgoing events on vertical seismic profiles. | ||

GB2309082B | 1999-12-01 | |||

GB2333364B | 2000-03-08 | |||

GB2337591B | 2000-07-12 | |||

GB2341680B | 2001-06-13 | |||

GB2358469B | 2002-02-27 |

Other References:

Aki et al Elastic waves from a point dislocation source Quantitative seismology—theory and mehods, W. H. Freeman & Co, 1980, pp. 68-69.

Amundsen Wavenumber-based filtering of marine point-source data Geophysics, vol. 58, No. 9, Sep. 1993, pp. 1335-1348, XP-002166869.

Amundsen et al Decomposition of multicomponent sea-floor data into upgoing and downgoing P- and S-waves Geophysics, vol. 60, No. 2, Mar.-Apr. 1995, pp. 563-572, XP-002130603.

Amundsen et al Multiple attenuation and P/S splitting of multicomponent OBC data at a heterogeneous sea floor Wave Motion, vol. 32, 2000, pp. 67-78.

Barr et al Attenuation of water-column reverberations using pressure and velocity detectors in a water-bottom cable Expanded abstracts from the 59th Ann. Internat. Mtg., Soc. Expl. Geophys., 1989, 653-656.

Dankbaar Separation of P- and S-waves Geophysical Prospecting, 33, 1985, pp. 970-986.

Holliger et al Effects of the shallow subsurface on upper crustal seismic reflection images Tectonophysics, 286, 1998, 161-169.

Osen et al Towards optimal spatial filters for de-multiple and P/S splitting of OBC data Expanded abstracts from the 68th Ann. Internat. Mtg., Soc. Expl. Geophys., 1998, 2036-2039.

Robertsson et al Viscoelastic finite-difference modeling Geophysics, vol. 59, No. 9, Sep. 1994, pp. 1444-1456, XP-002166870.

Robertsson et al Source-generated noise in shallow seismic data European J. of Env. and Eng. Geophys., vol. 1, 1996, pp. 107-124.

Robertsson et al Wavefield separation using a volume distribution of three component recordings Geophys. Res. Lett., vol. 26, No. 18, 1999, pp. 2821-2824.

Schalkwijk et al Decomposition of multicomponent ocean-bottom data in two steps Expanded abstracts from the 68th Ann. Internat. Mtg., Soc. Expl. Geophys., 1998, 1425-1428.

Spiegel Formulas from Vector analysis Mathematical handbook of formulas and tables, McGraw-Hill, 1968, p. 120.

White Plane waves Seismic waves: radiation, transmission and attenuation, McGraw-Hill, 1965, pp. 14-77.

Zhou et al Numerical seismogram computations for inhomogeneous media using a short, variable length convolutional differentiator Geophysical Prospecting, vol. 41, 1993, pp. 751-766, XP-000994570.

Amundsen Wavenumber-based filtering of marine point-source data Geophysics, vol. 58, No. 9, Sep. 1993, pp. 1335-1348, XP-002166869.

Amundsen et al Decomposition of multicomponent sea-floor data into upgoing and downgoing P- and S-waves Geophysics, vol. 60, No. 2, Mar.-Apr. 1995, pp. 563-572, XP-002130603.

Amundsen et al Multiple attenuation and P/S splitting of multicomponent OBC data at a heterogeneous sea floor Wave Motion, vol. 32, 2000, pp. 67-78.

Barr et al Attenuation of water-column reverberations using pressure and velocity detectors in a water-bottom cable Expanded abstracts from the 59th Ann. Internat. Mtg., Soc. Expl. Geophys., 1989, 653-656.

Dankbaar Separation of P- and S-waves Geophysical Prospecting, 33, 1985, pp. 970-986.

Holliger et al Effects of the shallow subsurface on upper crustal seismic reflection images Tectonophysics, 286, 1998, 161-169.

Osen et al Towards optimal spatial filters for de-multiple and P/S splitting of OBC data Expanded abstracts from the 68th Ann. Internat. Mtg., Soc. Expl. Geophys., 1998, 2036-2039.

Robertsson et al Viscoelastic finite-difference modeling Geophysics, vol. 59, No. 9, Sep. 1994, pp. 1444-1456, XP-002166870.

Robertsson et al Source-generated noise in shallow seismic data European J. of Env. and Eng. Geophys., vol. 1, 1996, pp. 107-124.

Robertsson et al Wavefield separation using a volume distribution of three component recordings Geophys. Res. Lett., vol. 26, No. 18, 1999, pp. 2821-2824.

Schalkwijk et al Decomposition of multicomponent ocean-bottom data in two steps Expanded abstracts from the 68th Ann. Internat. Mtg., Soc. Expl. Geophys., 1998, 1425-1428.

Spiegel Formulas from Vector analysis Mathematical handbook of formulas and tables, McGraw-Hill, 1968, p. 120.

White Plane waves Seismic waves: radiation, transmission and attenuation, McGraw-Hill, 1965, pp. 14-77.

Zhou et al Numerical seismogram computations for inhomogeneous media using a short, variable length convolutional differentiator Geophysical Prospecting, vol. 41, 1993, pp. 751-766, XP-000994570.

Primary Examiner:

Lobo, Ian J.

Attorney, Agent or Firm:

Wang, William L.

Batzer, William B.

Ryberg, John J.

Batzer, William B.

Ryberg, John J.

Claims:

What is claimed is:

1. A method of creating a filter for use with locally dense seismic data comprising the steps of: obtaining survey geometry characteristics from a locally dense seismic survey designed to record characteristics of an elastic or acoustic wavefield, the survey comprising a plurality of groups of receivers, and each group comprising at least three receivers densely spaced from each other; designing a filter which uses spatial derivatives of the wavefield of order between or including one and the maximum order of spatial derivatives of the wavefield that can be estimated within a group, such that the filter when combining data from within a single group, separates components of some or all of the wavefield arriving at the single group.

2. A method according to claim 1 wherein the filter is designed to separate up/down going components of some or all of the wavefield.

3. A method according to claim 1 wherein the filter is designed to separate p/s components of some or all of the wavefield.

4. A method according to claim 1 wherein the filter is designed to separate p/s and up/down components of some or all of the wavefield.

5. A method according to claim 1 wherein each of the densely spaced receivers in the group are spaced apart such that statics in the portion of the wavefield of interest are substantially constant.

6. A method according to claim 1 wherein each of the densely spaced receivers in the group are spaced apart by about 2 meters or less.

7. A method according to claim 1 wherein each of the densely spaced receivers in the group are spaced apart by a distance of about one fifth the shortest wavelength of interest or less.

8. A method according to claim 1 wherein the locally dense seismic survey comprises generating elastic or acoustic waves, and the receivers in a group span less than about the smallest wavelength of said elastic or acoustic waves.

9. A method according to claim 1 further comprising the step of calculating partial derivatives of the wavefield.

10. A method according to claim 9 wherein the step of calculating partial derivatives includes using a taylor series expansion as an approximation.

11. A method according to claim 9 further comprising the step of estimating near-surface material properties.

12. A method according to claim 11 further comprising the step of filtering the seismic data by combining the estimated near surface material properties, the seismic data, and the calculated partial derivatives.

13. A method according to claim 1 wherein a portion of the wave field that is separated is the pressure wavefield.

14. A method according to claim 13 wherein the filter separates surface waves and/or air wave induced ground motion from the seismic data.

15. A method according to claim 1 wherein the locally dense seismic survey is performed on land.

16. A method according to claim 15 wherein the step of creating the filter comprises using the free surface condition to convert vertical derivatives of the wavefield to horizontal derivatives of the wavefield.

17. A method according to claim 1 wherein the seismic survey is performed primarily for hydrocarbon reservoir exploration, evaluation or characterisation.

18. A method according to claim 1 wherein the near surface velocity is essentially isotropic.

19. A method according to claim 1 wherein the near surface velocity is anisotropic.

1. A method of creating a filter for use with locally dense seismic data comprising the steps of: obtaining survey geometry characteristics from a locally dense seismic survey designed to record characteristics of an elastic or acoustic wavefield, the survey comprising a plurality of groups of receivers, and each group comprising at least three receivers densely spaced from each other; designing a filter which uses spatial derivatives of the wavefield of order between or including one and the maximum order of spatial derivatives of the wavefield that can be estimated within a group, such that the filter when combining data from within a single group, separates components of some or all of the wavefield arriving at the single group.

2. A method according to claim 1 wherein the filter is designed to separate up/down going components of some or all of the wavefield.

3. A method according to claim 1 wherein the filter is designed to separate p/s components of some or all of the wavefield.

4. A method according to claim 1 wherein the filter is designed to separate p/s and up/down components of some or all of the wavefield.

5. A method according to claim 1 wherein each of the densely spaced receivers in the group are spaced apart such that statics in the portion of the wavefield of interest are substantially constant.

6. A method according to claim 1 wherein each of the densely spaced receivers in the group are spaced apart by about 2 meters or less.

7. A method according to claim 1 wherein each of the densely spaced receivers in the group are spaced apart by a distance of about one fifth the shortest wavelength of interest or less.

8. A method according to claim 1 wherein the locally dense seismic survey comprises generating elastic or acoustic waves, and the receivers in a group span less than about the smallest wavelength of said elastic or acoustic waves.

9. A method according to claim 1 further comprising the step of calculating partial derivatives of the wavefield.

10. A method according to claim 9 wherein the step of calculating partial derivatives includes using a taylor series expansion as an approximation.

11. A method according to claim 9 further comprising the step of estimating near-surface material properties.

12. A method according to claim 11 further comprising the step of filtering the seismic data by combining the estimated near surface material properties, the seismic data, and the calculated partial derivatives.

13. A method according to claim 1 wherein a portion of the wave field that is separated is the pressure wavefield.

14. A method according to claim 13 wherein the filter separates surface waves and/or air wave induced ground motion from the seismic data.

15. A method according to claim 1 wherein the locally dense seismic survey is performed on land.

16. A method according to claim 15 wherein the step of creating the filter comprises using the free surface condition to convert vertical derivatives of the wavefield to horizontal derivatives of the wavefield.

17. A method according to claim 1 wherein the seismic survey is performed primarily for hydrocarbon reservoir exploration, evaluation or characterisation.

18. A method according to claim 1 wherein the near surface velocity is essentially isotropic.

19. A method according to claim 1 wherein the near surface velocity is anisotropic.

Description:

The present invention relates to the field of seismic data acquisition and processing. In particular, the invention relates to a system and method for seismic wavefield separation.

Optimal processing, analysis and interpretation of land seismic data ideally require full information about the wavefield so that the wavefield can be separated into its up- and down-going and P- and S-components as well as determining phase and polarity. For 3C acquisition of land surface seismic data it is common practice simply to interpret the vertical component as the P-section and the horizontal components as SV- and SH-sections. This “traditional” P/S interpretation is exact for vertical arrivals. However, as energy is incident away from normal incidence angles, this approximation breaks down, both because of projections on to all components, but also because of a non-unity reflection coefficients and mode-conversions at the free-surface.

Exact analytical filter expressions for wavefield separation have previously been derived by for instance Dankbaar, J. W. M., 1985, Separation of P- and S-waves: Geophys. Prosp., 33, 970-986, and these have been applied to seismic data in conventional recording geometries. Unfortunately, statics problems severely limit the practical use of these wave-equation based techniques.

Thus, it is an object of the present invention to provide a filtering technique that lends itself to be applied within local densely deployed single sensor receiver groups where statics are substantially constant.

It is a further object of the present invention to provide a filtering technique that can be implemented efficiently directly in the spatial domain.

According to the invention a method of creating a filter for use with locally dense seismic data is provided. The method includes obtaining survey geometry characteristics from a locally dense seismic survey designed to record characteristics of an elastic or acoustic wavefield. The seismic survey is made up of a number of groups of receivers, with each group comprising at least three receivers densely spaced from each other. According to the method, a filter is designed which uses spatial derivatives of the wavefield of order between 1 and the maximum order of spatial derivatives of the wavefield that can be estimated within a group. The filter is designed such that when combined with data from within a single group, the filter separates components of some or all of the wavefield arriving at the single group.

The filter can be designed so as to separate up/down going components, p/s components, or both up/down and p/s components.

The densely spaced receivers in the group are preferably spaced apart such that statics in the portion of the wavefield of interest are substantially constant. More preferably, each of the densely spaced receivers in the group are spaced apart by about 2 meters or less, or by a distance of about one fifth the shortest wavelength of interest or less.

Partial derivatives of the wavefield are also preferably calculated, and this can be done using a taylor series expansion as an approximation. According to the invention, the seismic data is preferably filtered by combining estimated near surface material properties, the seismic data, and the calculated partial derivatives (both in space and time).

The filter can also be used to separate surface waves or airwave induced ground motion from the seismic data. The free surface condition can be used to convert vertical derivatives of the wavefield to horizontal derivatives of the wavefield.

According to the invention, the seismic survey is performed primarily for the purpose hydrocarbon reservoir exploration, evaluation or characterisation, although other uses can be made.

The invention is applicable where the near surface velocities are isotropic or anisotropic.

*a-c *

*a ***3***b *

*a ***4***b *

*a-d *_{3 }

*a-d *

According to the invention a new approach is provided for P/S separation of land surface seismic data and for removing the effects of the Earth's free surface (i.e. up/down separation). By converting vertical spatial derivatives to horizontal derivatives using the free-surface condition, the methodology can make use of locally dense measurements of the wavefield at the free surface to calculate all spatial derivatives of the wavefield. These can in turn be used to compute divergence (P-waves) and curl (S-waves) of the wavefield at the free surface.

The effects of the free surface are preferably removed through an up/down separation step using the elastodynamic representation theorem. This results in expressions where spatial filters are convolved with recorded data. The filters can be successfully approximated so that they fit the local dense acquisition pattern used for the P/S separation step. In particular, the simplest approximate expression for up-going P-waves consists of two terms. The first term preferably corresponds to the divergence in the presence of the free surface scaled by a material constant. The second term preferably is a time derivative of the recorded vertical component scaled by a material constant. Hence, a correction is added to the “traditional” P-interpretation through the first term, which improves accuracy for incidence angles outside normal incidence.

In UK Patent Application entitled “System and Method for Estimating Seismic Material Properties” (UK Patent Application No. 0003410.8) filed on 15 Feb. 2000, incorporated herein by reference, a method is provided than can use the volumetric recordings of the wavefield to invert for P and S velocities in the Earth in the neighbourhood of a small, closely-spaced array of receivers. “Volumetric recording” refers to an array that approximately encloses a volume of the Earth. The quantities estimated are the effective velocities of the P- and S-components of the wavefield at any point in time. Hence, these will vary with both wave type and wavelength. If estimated for the near surface Earth structure, such velocities may be useful for statics estimation, or for separation of the wavefield into up- and down-going components as described herein.

Implications of the free surface condition for land seismic recording will be discussed first with reference to the fully anisotropic case. The elastic constitutive relation relates components of the stress tensor σ_{ij }_{ij}

_{ij}*=c*_{ijkl}_{kl}

where c_{ijkl }_{1 }_{2 }_{3}

where C is the symmetric stiffness matrix with 21 independent components:

The strain ε_{ij }_{i }

where ∂i denotes spatial derivatives in the x_{1}_{2 }_{3 }

The free-surface condition at the Earth's surface gives us three additional constraints:

_{i3}

which are sufficient to allow us to compute the remaining velocities given that we make some independent estimate of the relevant elastic stiffnesses.

In addition, constraints on the relation between individual elements of the stress and strain tensors could possibly be used to correct for coupling or to compute near-surface properties.

The case of isotropic material properties in the near-surface environment is of particular interest in the land surface seismic case. Using the Voigt notation, the stiffness matrix takes the following form:

where λ and μ are the Lamé constants.

The constraints imposed by the free-surface condition (5) become: _{3}_{2}_{2}_{3}

_{3}_{1}_{1}_{3}

where the horizontal derivatives on the right hand side are known from the surface measurements. Note that the material properties only occur in equation (7) and not in equations (8) and (9).

Divergence and curl of a wavefield at a free surface overlaying a homogeneous isotropic half space will now be discussed. This discussion is in the context of an isotropic media. The elastic wave equation for particle displacement u can be written as:

*ü=f+**∇·u**∇×u*

where ρ is the density and f denotes a distribution of body forces. Lamé's theorem states that there exist potentials Φ and Ψ of u with the following properties:

*u=∇Φ+∇×Ψ,*

where c_{α}_{β }

*·u=∇*^{2}

*∇×u=∇×∇×Ψ*

By measuring the curl and the divergence of an elastic wavefield we can thus measure the P- and S-wave components separately.

An acquisition pattern comprising tetrahedra of 3C measurments can be used to achieve the separation of the wavefield into its curl- and divergence-free components. See, e.g. Robertsson, J. O. A., and Muyzert, E., 1999, Wavefield separation using a volume distribution of three component recordings: Geophys. Res. Lett., vol. 26, 2821-2824, incorporated herein by reference. Such acquisition patterns are described in further detail in UK Patent Application No. 9921816.6, incorporated herein by reference. All spatial derivatives of the wavefield components can be calculated. Therefore, divergence and curl can be calculated explicitly from surface measurements only. Equations (7), (8) and (9) give us the following expressions for divergence and curl of particle velocity at a free surface: *∇×v*_{1}_{2}_{3}

*∇×v*_{2}_{1}_{3}

*∇×v*_{3}_{1}_{2}_{2}_{1}

First we note that the ratio

(may be frequency dependent) scales the expression for the divergence in equation (17).

Second, equations (17), (18), (19) and (20) contain both the up-going and down-going parts of the wavefield. This includes mode conversions at the free surface. For instance, the divergence given by (17) contains not only the desired up-going P-waves, but also, the down-going P-to-P reflection, down-going S-to-P conversions, etc. Moreover, a plane P-wave which is vertically incident on the free-surface will have zero divergence (up- and down-going parts interfere destructively). Removing the effects of the free surface is therefore regarded as an important step in the preferred P/S separation technique.

A technique for Up/down wavefield separation using the elastodynamic representation theorem will now be discussed. The elastodynamic representation theorem or Betti's relation can be derived from the equation of motion and the elastic constitutive relations using Gauss' theorem. Suppose that we have a volume V enclosed by a surface S, and that we wish to calculate the displacement of a wavefield u at a point x′ in V. The displacement u(x′) is directly related to the stress σ and displacement along S, sources in V, as well as the displacement Green's tensor G_{ij }_{ijk }

where {circumflex over (n)} is the normal unit vector to S, t_{i}_{i }^{th }

*t={circumflex over (n)}·σ.*

Now, suppose that the volume V consists of a homogeneous elastic medium. *a-c *_{1}_{2 }*a*_{1 }_{2 }

The expression in equation (23) represents an up-going wavefield at x′ since the field is up-going at S_{1}

Next consider the situation depicted in *b. *_{1 }

Finally, we consider the situation shown in *c. *_{1 }

In equation (24) we have used particle velocities ν_{i }_{i }_{1 }_{i3}

Hereinafter the tilde denotes a wavefield with up-going waves only, i.e. having removed the effects of the free surface from the wavefield.

According to a preferred embodiment of the invention, the elastodynamic representation theorem can be used to extract the desired up-going wavefield from surface seismic recordings. For a discussion in the case of seismic data from ocean bottom cables, see Published UK Patent Application GB 2 333 364 A, incorporated herein by reference.

In the following discussion, ω is the angular frequency, k_{α}_{α }_{β}_{β }_{α}_{β }

The elastic displacement Green's tensor in an isotropic homogeneous medium is:

Where g_{α }_{β}_{in }

The corresponding isotropic stress Green's tensor is given in terms of the displacement Green's tensor as:

_{ijk}_{ij}_{k}*G*_{kn}_{j}*G*_{in}_{i}*G*_{jn}

A flat horizontal free surface is assumed here. Expressions are therefore needed for Green's functions such that x=(ξ, x_{3}_{3}_{3}_{3}_{3}_{∈}^{+}_{1 }_{2 }

Since the “free-space” Green's tensors G_{kl }_{ijk }

The filters in equations (29), (30), and (31) are:

*F*_{ν}_{3}^{ν}^{ν}_{33ν}*x*_{3}*=x*_{3}^{‘}_{γ}*k*_{β}^{−2}_{ζ}_{ζ}*g*_{α}*k*_{β}^{−2}_{3}^{2}*g*_{β}

*F*_{ν}_{ν}^{ν}^{3}_{3ν3}*x*_{3}*=x*_{3}^{‘}*k*_{β}^{−2}_{ν}_{3}^{2}*g*_{α}*k*_{β}^{2}_{3}^{2}*g*_{β}

where the subscripts ν and ζ denote either index 1 or 2 corresponding to horizontal coordinates x_{1 }_{2}

Using the Green's function in equation (27), we can obtain explicit expressions for the filters in equations (32) and (33) that can be implemented directly. In the fk-domain these are:

where k_{3}^{(α)}_{α}^{2}_{ζ}_{ζ}_{3}^{(β)}_{β}^{2}_{ζ}_{ζ}

A preferred technique for P/S separation of surface seismic data will now be discussed. We can calculate the divergence and curl of the up-going waves by taking spatial derivatives of equations (29), (30) and (31). Some care must be taken here since this involves spatial derivatives of the stress Green's tensor Σ_{ijk}

*×{circumflex over (v)}*_{1}_{2}_{3}*+F*_{υ1}^{(∇×v)}^{1}*x*_{1}*x**F*_{ν2}^{(∇×v)}^{1}*x*_{2}*x*

*×{circumflex over (v)}*_{2}_{1}_{3}*+F*_{υ1}^{(∇×v)}^{2}*x*_{1}*x**F*_{υ2}^{(∇×v)}^{2}*x*_{2}*x*

Again, using the Green's function in equation (27), we can obtain explicit expressions for the filters in equations (36), (37) and (38). In the fk-domain these are: *F*_{ν2}^{(∇×v)}^{2}*=−F*_{ν1}^{(∇×v)}^{1}

The traditional way for P-wave interpretation is to simply look at the recorded ν_{3 }_{3 }_{3 }

Equations (40) to (44) are all functions of the material properties c_{α }_{β}

An example of a preferred implementation of wavefield separation filters will now be discussed. The filters derived above can in theory be implemented directly which yield expressions that are exact for homogeneous media with a flat surface. These expressions would remove all down-going and evanescent wave types including ground-roll.

However, the filters are slowly decaying and contain some complicating factors. High-order factors of k_{1 }_{2 }_{3}^{(α) }_{3}^{(β)}_{α}_{β}

One straightforward filter approximation is to make Taylor expansions around k_{ζ}_{ζ}

The application of a preferred wavefield separation approach to synthetic data will now be discussed. A reflectivity code was used to test the wavefield separation approach according to a preferred embodiment of the invention. See e.g., Kennett, B. L., 1983, Seismic wave propagation in stratified media: Cambridge University Press, Cambridge. This was chosen as opposed to for instance finite differences since up- and down-going wavefields can be calculated separately. Moreover, the quantities right below an interface can be obtained accurately. The output from the reflectivity code are particle velocities and divergence of particle displacement.

*a ***3***b **a *

As shown in *a ***202****204****206****208****210***a ***212****204**

*b ***210****210****250****252****254****256****250****254****252****256***b ***260****206****208**

Advantages of moving to smaller inter-receiver spacings include increased accuracy in the estimated derivatives, and greater validity in the assumption that the material properties are the same in the vicinity of the receivers. However, an advantage of wider spacing is less sensitivity to noise. These competing concerns in combination with the wavelength of elastic or acoustic waves (or more accurately, the projection of the wavefield on the recording surface), should be taken into consideration when designing the receiver group spacing. According to a presently preferred embodiment, the locally dense receivers are spaced about 1 meter apart. However spacing can in some situations be larger, for e.g. around 2 meters, or smaller, e.g. 0.5 meters. According to a preferred embodiment the inter-receiver spacing is around 0.25 meters or less. As mentioned, the optimal spacing of the receivers depends upon the wavelength of interest. There should be at least two receivers within the projection of the shortest wavelength of interest on the recording surface. According to a preferred embodiment the receivers are spaced apart by a distance approximately equal to or less than one-fifth of the shortest wavelength of interest.

In the model based on the geometry shown in *a ***3***b, *

*a ***4***b **a **b **a. *

Traditionally, 3C data have been interpreted by assuming that waves propagate vertically near the receivers (steep gradients in material properties are assumed in the near-surface region). Hence, P-waves show up on the vertical v_{3 }_{1 }_{2 }*a-d *_{3 }_{3 }*a *_{3 }*b *

Part of the problem is of course that the v_{3 }_{1 }_{2 }_{3 }*c **d *_{3 }

*a-d **a *_{1 }_{2 }_{3}*b *

In *c *_{1 }_{2 }_{3}*d *

By comparing all the difference sections in

According to the invention a new approach for P/S separation of land surface seismic data and removing the effects of the free surface is provided. By converting vertical derivatives to horizontal derivatives through the use of the free surface condition, the methodology can be used even when measurements are taken only at the free surface. Therefore by making locally dense measurements of the wavefield, all spatial derivatives needed to compute divergence and curl of the wavefield at the free surface can be obtained. These in turn correspond to P- and S-waves in isotropic media.

The effects of the free surface can be removed through an up/down separation step as described herein. The filter for P-waves depends on both P- and S-velocity at the receivers, whereas the S-wave filters only depend on the S-velocity. Approximations to the filters were derived using Taylor approximations and tested on synthetic data.

Filters have been derived for up/down and P/S separation using plane wave expansions in Dankbaar, J. W. M., 1985, Separation of P- and S-waves: Geophys. Prosp., 33, 970-986. These expressions can be compared to our expressions for full filters. A principal difference between the work by Dankbaar and those described herein, is that by deploying dense configurations of 3C geophones, P/S and up/down separation in 3D for each recording station can be done separately. The preferred up/down separation step makes use of approximations to spatial filters to obtain operators that are consistent with the number of geophones in the recording station. This approach is more robust since statics and near surface properties should be consistent within each recording station.

The present invention can also be implemented in several steps where for instance the P/S separation is performed in 3D. The up/down separation can be performed in 3D or in 2D using an implicit filter if data are acquired along a 2D line for instance.

For 3C acquisition of land surface seismic data it is common practice simply to interpret the vertical component as the P-section and the horizontal components as SV- and SH-sections. This “traditional” P/S interpretation is exact for vertical arrivals. However, as energy is incident at non-normal incidence angles, this approximation tends to break down, both because the different waves appear on all components, but also because reflection coefficients differ from unity and mode-conversions occur at the free-surface. By comparing the “traditional” P-sections to the new methodology using synthetic data, we find a significant improvement in obtaining accurate amplitude and phases of arrivals for non-normal incidence angles. By simply using a zeroth-order Taylor approximation, we obtained sufficiently accurate results up to incidence angles of up to around 30° away from normal incidence (this result depends on material properties in the example). Note that a zeroth-order Taylor approximation only involves first-order derivatives in time and space (along the free surface). Note that the approximate expression for divergence consists of two terms. The first term corresponds to the divergence in the presence of the free surface scaled by a material constant. The second term is a time derivative of v_{3 }

**150****152****154****166****166****170****162****164****156****158****160****128***b. *

Importantly, according to the preferred embodiment, the spacing of between the receivers within a single receiver group is substantially less than the spacing between the receiver groups. Schematically, this is shown in **122****160****120****160****158***a ***3***b, ***120****122**

Referring again to **156****158****160****170****170****170****176****180****180****346****352****8**

**340***a, ***3***b, ***7**

In step **342**

In step **346****340**

In step **344**

In step **350****346**

In step **348**

In step **352****346****348****344****350****356**

While preferred embodiments of the invention have been described, the descriptions are merely illustrative and are not intended to limit the present invention. For example, while the preferred embodiments of the invention have been described primarily for use on the land surface, the invention is also applicable to receivers placed on and below the ocean floor. In the case of ocean bottom receivers, it is preferable to use stress conditions relevant for fluid-solid boundaries rather than the free surface condition. Additionally, the present invention is applicable to seismic measurements made in a borehole, known as borehole seismics. Although the examples described assume an essentially isotropic medium in the near surface region, the invention is also applicable to anisotropic media. In the case of anisotropic media, one may wish to increase the number geophones per group.